Question

$x^x$ has a stationary point at
[Karnataka CET 1993]

• A. x = e
• B. $x=\frac{1}{e}$
• C. x = 1
• D. $x=\sqrt{e}$

Answer 1

The correct option is B $x=\frac{1}{e}$

$tanu=x+y=x(1+\frac{y}{x})$

$\therefore$ tan u is homogeneous in x,y of order 1.
$\therefore x\frac{\partial }{\partial y}\left(tanu\right)=tanu$
$\therefore se{c}^{2}u\frac{\partial }{\partial x}+yse{c}^{2}u\frac{\partial u}{\partial y}=tanu$
$\therefore x\frac{\partial u}{\partial x}+yy\frac{\partial u}{\partial y}=tanu.co{s}^{2}u=sinucosu=\frac{1}{2}sin2u.$